A topology-preserving deformation method for three-dimensional models based on multi-body harmonic fields

Abstract

The present invention discloses a topology-preserving deformation method for a three-dimensional model based on a multi-body harmonic field, and belongs to the fields of computer graphics, computational mathematics, topology, and differential geometry. First, a tetrahedral mesh is constructed between the original three-dimensional model and the target three-dimensional model, and then the region where the topological transformation occurs during the deformation process is calculated based on the traditional one-body harmonic field, and a special multi-boundary condition is set, and then the multi-body harmonic field is calculated based on this, and finally a topology-preserving surface deformation is induced. The present invention can find the region where the topological transformation occurs during the deformation process based on the saddle point in the traditional body harmonic field, and adaptively construct a multi-body harmonic field that can induce topology-preserving deformation, thereby generating a topology-preserving deformation surface under the guidance of the multi-body harmonic field. The method is universal and efficient for the deformation of three-dimensional models with the same topology, requires less computational cost than the traditional large deformation differential homeomorphism metric mapping, and can be widely used.

Description

Technical Field

[0001] The present invention belongs to the fields of computer graphics, computational mathematics, topology, and differential geometry, and relates to a method for constructing a three-dimensional differential homeomorphism mapping based on a multi-volume harmonic field, which is applicable to low-genus surfaces with complex geometry. The method uses a multi-volume harmonic field as a guide field to calculate anisotropic volume harmonic fields, thereby constructing a three-dimensional differential homeomorphism mapping. Background Art

[0002] In the two-dimensional case, the harmonic mapping of two convex plane regions is diffeomorphic if and only if the boundary conditions are diffeomorphic; however, in the three-dimensional case, the construction of diffeomorphic mapping is difficult. Currently, the only universal method is the large deformation diffeomorphic metric mapping (LDDMM), but its computational cost is quite expensive. Diffeomorphic mapping plays a very important role in computer graphics, medical image analysis, and finite element mesh generation. Three-dimensional diffeomorphic mapping can induce topology-preserving surface deformation in space, which is a powerful tool for the generation of boundary layer meshes required in computational fluid dynamics, and can solve the mesh generation of grooves and concave cavities that traditional methods cannot apply. Summary of the invention

[0003] Based on the above problems, the present invention proposes a method for constructing three-dimensional differential homeomorphism mapping based on multi-body harmonic fields, which includes four invention contents:

[0004] 1. Tetrahedral discrete mesh generation of the deformation space between 3D models;

[0005] 2. The construction of multi-body harmonic fields;

[0006] 3. Generation of anisotropic scalar fields guided by multi-body harmonic fields;

[0007] 4. Topology-preserving deformation of three-dimensional models based on anisotropic scalar fields.

[0008] The technical solution of the present invention:

[0009] 1. Generation of tetrahedral discrete meshes in the differential homeomorphic deformation space between the original model and the surrounding model

[0010] 1) Input the original 3D model and its surrounding 3D model with the same topology, usually using a triangular surface mesh or a quadrilateral surface mesh;

[0011] 2) The deformation space of the three-dimensional model is defined as the space between the original three-dimensional model and the surrounding three-dimensional model;

[0012] 3) Use tetrahedral meshes to discretize the deformation space. If there is a tetrahedral mesh unit whose four vertices are on the original 3D model or the surrounding 3D model, split the tetrahedral mesh unit and repeat until there is no such situation;

[0013] 4) Subdivide the deformed space grid according to the model characteristics. First, use the Laplace operator to calculate the scalar field in the tetrahedral grid and calculate the position of the saddle point (singular point) in the scalar field. Then, subdivide the tetrahedral grid cells near the saddle point (singular point) to improve the local grid resolution. Finally, optimize the locally subdivided tetrahedral grid to improve the quality of the grid cells.

[0014] 2. Construction of multi-body harmonic field

[0015] 1) Definition of multi-body harmonic field on the vertices of tetrahedral mesh:

[0016] H(v p )=[H 1 (v p ),H 2 (v p ),...,H d (v p )], (1)

[0017] Among them, d is the multi-index of the multi-body harmonic field, H i (v p ) is the i-th harmonic field acting on the tetrahedral mesh vertex v p The scalar value obtained above, i = 1, 2, … d.

[0018] 2) The definition of Laplace equation for calculating multi-body harmonic field on tetrahedral grid:

[0019] LG=0, (2)

[0020] Among them, L is the weight matrix for solving the Laplace equation, and the expression is:

[0021]

[0022] Among them, e pq are the edges on the tetrahedral mesh, whose endpoints are v p With v q ,N(v p ) is v p The neighborhood point set, G is the multi-body harmonic field matrix, and the matrix expression is:

[0023]

[0024] w(e pq) is used to calculate the edge weight of the Laplace equation. Here, the classic cotangent weight is used, and the expression is:

[0025]

[0026] Among them, N(e pq ) is e pq The tetrahedral grid cell neighborhood, l ij ,θ ij e ij The side length of the tetrahedral mesh element and the ij The dihedral angle, e ij The tetrahedral mesh elements connected except v p With v q The edges of the other two vertices.

[0027] 3) Multi-volume harmonic field calculation on tetrahedral mesh:

[0028] The multi-body harmonic energy T(H) is defined as:

[0029]

[0030] Where E is the edge set on the tetrahedral mesh.

[0031] The calculation of multi-body harmonic field on tetrahedral grid is transformed into the optimization of multi-body harmonic energy T(H). When T(H) is minimized, H is the multi-body harmonic field.

[0032] The present invention uses an iterative method to optimize the multi-body harmonic energy T(H), and the iterative formula is:

[0033]

[0034] The iterative calculation steps of the multi-body harmonic field are as follows:

[0035] Set d-fold Dirichlet boundary conditions H(v) on the original 3D model and the surrounding model. l )=C, where v l is the control point, C is a d-dimensional constant; the maximum number of iterations for optimizing the multi-body harmonic energy is set to k iter (generally set to 2000), the energy optimization cutoff threshold is t energy (generally 1e-6); iteratively update the value of the multi-body harmonic field acting on the tetrahedral mesh vertices based on formula (7) until the energy cutoff threshold t is met energy Or reach the maximum number of iterations k iter .

[0036] 4) Construction of boundary conditions for multi-body harmonic fields that are suitable for inducing topological deformation of three-dimensional models:

[0037] a. Construct a one-dimensional harmonic field in the deformation space. Set the energy of the original 3D model to a scalar constant x (usually 1), the energy of the target 3D model to a scalar constant y (usually 0), and calculate the one-dimensional harmonic field on the tetrahedral mesh;

[0038] b. Calculate the saddle point (singular point) in the harmonic field (scalar field) of a body, denoted by S = {s 1 ,s 2 ,…s n}. The saddle point (singular point) provides the location where the topological change occurs when the original three-dimensional model is deformed to the target three-dimensional model under the induction of the volume harmonic field. If there is no saddle point (singular point) in the volume harmonic field (scalar field), there is a topological deformation that can be induced directly from the original three-dimensional model to the target three-dimensional model based on the isosurface;

[0039] c. Extraction of boundary constraint points of multi-body harmonic field. According to the saddle point, along the harmonic field gradient, we can detect two or more corresponding feature points or feature regions on the model. From the saddle point (singular point) si of the harmonic field of a multi-body, we can trace back to the three-dimensional model along the gradient line. Generally, we can trace back to at least two vertices or two regions, which are recorded as

[0040] d. Based on the extracted feature point (region) pairs, each feature point pair is used as the corresponding Dirichlet boundary condition to construct a volume harmonic field with corresponding multiple indices (number of feature point pairs). If there are n saddle points (singular points) in the volume harmonic field (scalar field), and each saddle point (singular point) is s i Corresponding to m i If there are traceable vertices or regions, you need to build Heavy body harmonizing field;

[0041] s i The corresponding j-th multi-body harmonic field The Dirichlet boundary condition is set as:

[0042]

[0043] Among them, α is generally taken as 1, and β is generally taken as 0.

[0044] 3. Generation of anisotropic scalar fields guided by multi-body harmonic fields. The details are as follows:

[0045] 1) Calculation of edge weights based on the anisotropy of multi-body harmonic fields:

[0046]

[0047] Among them, λ is a parameter that controls the influence of the multi-body harmonic field on the edge weight (λ is generally set to 10): the smaller λ is, the smaller the edge weight is. The smaller the influence of the multi-body harmonic field H is, the larger λ is, and the edge weight The greater the influence of the multi-body harmonic field H.

[0048] 2) Scalar field calculation based on anisotropic edge weights:

[0049]

[0050] Where F is the anisotropic scalar field acting on the tetrahedral mesh, is the weighted Laplace matrix, expressed as:

[0051]

[0052] 3) Setting of Dirichlet boundary conditions:

[0053]

[0054] Among them, γ and χ are constant scalars (γ is generally 1; χ is generally 0); M source , M target They are the original 3D model and the target 3D model respectively.

[0055] 4. Topology-preserving deformation of 3D models based on anisotropic scalar fields

[0056] 1) Extract n isosurfaces from the anisotropic scalar field and denote them as g = {g 1 ,g 2 ,…,g n}, each isosurface is a state of the deformation process.

[0057] 2) From point v on the original model k ∈M source Starting from the gradient of the anisotropic scalar field, trace back to n isosurfaces and obtain n new vertices respectively.

[0058] 3) According to the new vertices obtained in each layer Using M-based source The vertex connection method can generate n topology-preserving deformation models {M 1 , M 2 , …, M n}.

[0059] Beneficial effects of the present invention: The method of topology-preserving deformation of three-dimensional models based on multi-volume harmonic fields proposed in the present invention can construct three-dimensional differential homeomorphism mappings and induce a family of topology-preserving deformation surfaces. In traditional surface deformation, if the original surface contains grooves or slit features, topological transformation often occurs during the deformation process, which is illegal in some specific applications, such as boundary layer mesh generation. Therefore, it is very meaningful and challenging to find a family of topology-preserving deformation surfaces between the original three-dimensional model and the target three-dimensional model. The present invention can find the area where topological transformation occurs during the deformation process based on the saddle point in the traditional volume harmonic field, and adaptively construct a multi-volume harmonic field that can induce topology-preserving deformation, thereby generating a topology-preserving deformation surface under the guidance of the multi-volume harmonic field. This method is universal and efficient for the deformation of three-dimensional models with the same topology, requires less computational cost than the traditional large deformation differential homeomorphism metric mapping (LDDMM), and can be more widely used. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is the algorithm flow chart of the present invention;

[0061] Figure 2 It is the original double-antenna three-dimensional model;

[0062] Figure 3 is the target cube three-dimensional model;

[0063] Figure 4 Saddle points and their traceable regions for the harmonic fields of deformed space bodies;

[0064] Figure 5 To reconcile the field-induced surface deformation for the traditional body;

[0065] Figure 6 Harmonize field-induced topology-preserving deformations of surfaces for multi-volumes. DETAILED DESCRIPTION

[0066] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and technical solutions.

[0067] The algorithm execution process of the present invention is divided into five calculation steps, such as Figure 1 :

[0068] 1) constructing a tetrahedral mesh between the original 3D model and the target 3D model;

[0069] 2) Detect the region where topological transformation occurs during deformation based on the traditional single-body harmonic field;

[0070] 3) Setting special multi-boundary conditions based on the saddle point of the traditional single-body harmonic field and calculating the multi-body harmonic field;

[0071] 4) Calculate anisotropic scalar fields based on multi-body harmonic fields;

[0072] 5) Construct topology-preserving deformable surfaces in deformation space based on anisotropic scalar fields.

[0073] The algorithm input of the present invention is:

[0074] 1) Original 3D model (usually a triangular surface mesh or a quadrilateral surface mesh);

[0075] 2) Target 3D model (usually a triangular surface mesh or a quadrilateral surface mesh);

[0076] 3) Number of deformations n.

[0077] The implementation case of the present invention takes the topology-preserving deformation of a two-antenna three-dimensional model to a cubic three-dimensional model as an example. At present, based on the existing methods, it is difficult to calculate the topology-preserving deformation of a two-antenna three-dimensional model to a cubic three-dimensional model in three-dimensional space. Figure 5 As shown, this example has illustrative value. The method based on multi-body harmonic field of the present invention can be well applied to such models and has scalability, such as Figure 6 As shown, the specific implementation steps are as follows:

[0078] 1. Input the double-antenna triangle surface mesh as the original 3D model, such as Figure 2 As shown; input the cube triangular surface mesh as the target 3D model, such as Figure 3 As shown; the input deformation times is 4;

[0079] 2. Use Tetgen tetrahedral mesh generation software to fill the tetrahedral mesh for the deformed space between the original 3D model and the target 3D model;

[0080] 3. Set the vertex energy of the original 3D model to 1 and the vertex energy of the target 3D model to 0 as boundary conditions, and calculate the traditional single-body harmonic field on the tetrahedral background grid;

[0081] 4. Calculate the saddle point S based on the traditional single-body harmonic field 1 , and trace along the gradient in the direction of increasing energy until the original three-dimensional model surface mesh, and obtain the traced area like Figure 4 As shown;

[0082] 5. The energy of the region is 1. The energy of the region is 0 and The energy of the region is 1. The energy of the region is 0 as the Dirichlet boundary condition, and the doublet harmonic field H = [H1 ,H 2 ];

[0083] 6. Calculate the anisotropic scalar field based on the multi-body harmonic field according to formula (9);

[0084] 7. According to the input deformation times 4, calculate the sampling energy g = {0.2, 0.4, 0.6, 0.8};

[0085] 8. Calculate the isosurface in the anisotropic scalar field based on the sampled energy and use it to construct the deformed surface {M 1 ,M 2 ,M 3 ,M 4},like Figure 6 shown.

Claims

1. A three-dimensional model topology-preserving deformation method based on multi-body harmonic field, characterized in that: Here are the steps: (1) Generation of tetrahedral discrete meshes in the diffeomorphic deformation space between the original model and the surrounding model (1.1) Input the original 3D model and its surrounding 3D model with the same topology, using a triangular surface mesh or a quadrilateral surface mesh; (1.2) The deformation space of the three-dimensional model is defined as the space between the original three-dimensional model and the surrounding three-dimensional model; (1.3) Discretize the deformation space using tetrahedral meshes; (1.4) Subdivide the deformed space grid according to the model characteristics; first, use the Laplace operator to calculate the scalar field in the tetrahedral grid and calculate the position of the saddle point in the scalar field; then, subdivide the tetrahedral units near the saddle point to improve the local grid resolution; finally, optimize the locally subdivided tetrahedral grid to improve the grid unit quality; (2) Construction of multi-body harmonic fields (2.1) Definition of multi-body harmonic field on the vertices of tetrahedral mesh: H(in p )=[H1(in p ),H2(in p ),...,H d (in p )], (1) Among them, d is the multi-index of the multi-body harmonic field, H i (v p ) is the i-th harmonic field acting on the tetrahedral mesh vertex v p The scalar value obtained above, i = 1, 2, ... d; (2.2) The definition of the Laplace equation for calculating the multi-body harmonic field on a tetrahedral grid is: LG=0, (2) Among them, L is the weight matrix for solving the Laplace equation, and the expression is: Among them, e pq are the edges on the tetrahedral mesh, whose endpoints are v p With v q ,N(v p ) is v p The neighborhood point set, w(e pq ) is the edge weight for calculating the Laplace equation, G is the multi-body harmonic field matrix, and the matrix expression is: (2.3) Multi-volume harmonic field calculation on tetrahedral grid: The multi-body harmonic energy T(H) is defined as: Where E is the edge set on the tetrahedral mesh; The calculation of multi-body harmonic field on tetrahedral grid is transformed into the optimization of multi-body harmonic energy T(H). When T(H) is minimized, H is the multi-body harmonic field. The iterative method is used to optimize the multi-body harmonic energy T(H), and the iterative formula is: (2.4) Construction of boundary conditions for multi-body harmonic fields that are suitable for inducing topological deformation of three-dimensional models: (2.4.1) Construct a harmonic field of a deformed space; (2.4.2) Calculate the saddle points in the one-dimensional harmonic field; the saddle points provide the locations where topological changes occur when the original three-dimensional model is deformed to the target three-dimensional model under the induction of the volume harmonic field; if there are no saddle points in the one-dimensional harmonic field, there is a topologically preserving deformation that can be induced directly based on the isosurface to the original three-dimensional model to the target three-dimensional model; (2.4.3) Extraction of boundary constraint points of multi-body harmonic fields; according to the saddle point, more than two corresponding feature points or feature regions on the model can be detected along the harmonic field gradient; (2.4.4) Based on the extracted feature point pairs, each feature point pair is used as the corresponding Dirichlet boundary condition to construct the corresponding multi-index, i.e., the volume harmonic field of the number of feature point pairs; (3) Generation of anisotropic scalar fields guided by multi-body harmonic fields. The details are as follows: (3.1) Calculation of edge weights based on the anisotropy of multi-body harmonic fields: Among them, λ is a parameter that controls the influence of the multi-body harmonic field on the edge weight: (3.2) Scalar field calculation based on anisotropic edge weights: Where F is the anisotropic scalar field acting on the tetrahedral mesh, is the weighted Laplace matrix, expressed as: (3.3) Setting of Dirichlet boundary conditions: Among them, γ and χ are constant scalars; M source , M target They are the original 3D model and the target 3D model respectively; (4) Topology-preserving deformation of three-dimensional models based on anisotropic scalar fields (4.1) In the anisotropic scalar field, n isosurfaces are extracted and denoted as g = {g1, g2, …, g n }, each isosurface is a state of the deformation process; (4.2) From the point v on the original model k ∈M source Starting from the gradient of the anisotropic scalar field, trace back to n isosurfaces and obtain n new vertices respectively. (4.3) According to the newly obtained vertices in each layer Using M-based source The vertex connection method can generate n topology-preserving deformation models {M1,M2,…,M n }.

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